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Chapter 14: Problem 50

What is a complete graph?

Short Answer

Expert verified
A complete graph is a graph in which every pair of vertices is connected by a line. Often denoted as \(K_n\), where \(n\) stands for the number of vertices in the graph. The number of edges in a complete graph with \(n\) vertices can be found using the formula \(\frac{n(n-1)}{2}\).

Step by step solution

01

Definition

A complete graph, often simply called a complete, is a type of graph in which every pair of graph vertices is connected by an edge. In other words, any two vertices in a complete graph are directly adjacent.
02

Notation and Examples

This concept can be denoted by the symbol \(K_n\), where the letter \(K\) represents a complete graph and \(n\) is the number of vertices of the graph. For instance, \(K_3\) is a complete graph with 3 vertices. It is often represented visually as a triangle, where each vertex is connected to every other vertex by a single edge.
03

Properties

A complete graph with \(n\) vertices has \(\frac{n(n-1)}{2}\) edges. For instance, a \(K_4\) complete graph with 4 vertices has 6 edges, while a \(K_5\) graph with 5 vertices has 10 edges.

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Most popular questions from this chapter

In Exercises 45-47, you have three errands to run around town, although in no particular order. You plan to start and end at home. You must go to the bank, the post office, and the market. Distances, in miles, between any two of these locations are given in the table. $$ \begin{aligned} &\text { DISTANCES (IN MILES) BETWEEN LOCATIONS }\\\ &\begin{array}{|l|c|c|c|c|} \hline & \text { Home } & \text { Bank } & \text { Post Office } & \text { Market } \\ \hline \text { Home } & * & 3 & 5.5 & 3.5 \\ \hline \text { Bank } & 3 & * & 4 & 5 \\ \hline \text { Post Office } & 5.5 & 4 & * & 4.5 \\ \hline \text { Market } & 3.5 & 5 & 4.5 & * \\ \hline \end{array} \end{aligned} $$ Use the Brute Force Method to find the shortest route to run your errands and return home. What is the minimum distance you can travel?

How do you determine if a graph has at least one Euler path, but no Euler circuit?

In Exercises 13-18, a connected graph is described. Determine whether the graph has an Euler path (but not an Euler circuit), an Euler circuit, or neither an Euler path nor an Euler circuit. Explain your answer. The graph has 77 even vertices and four odd vertices.

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Use a tree to model the parent-child relationships in the following family: Peter has three children: Zoila, Keanu, and Sandra. Zoila has two children: Sean and Helen. Keanu has no children. Sandra has one child: Martin. Use vertices to model the people and edges to represent the parent-child relationships.
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